(navigation image)
Home American Libraries | Canadian Libraries | Universal Library | Community Texts | Project Gutenberg | Children's Library | Biodiversity Heritage Library | Additional Collections
Search: Advanced Search
Anonymous User (login or join us)
Upload
See other formats

Full text of "Scientific Papers - Vi"

377.
ON THE STABILITY OF THE LAMINAR MOTION OF AN INVISCID FLUID.
[Philosophical Magazine, Vol. xxvi. pp. 10011010, 1913.]
THE equations of motion of an inviscid fluid are satisfied by a motion such that U, the velocity parallel to x, is an. arbitrary function of y only, while the other component velocities V and W vanish. The motion may be supposed to be limited by two fixed plane walls for each of which y has a constant value. In order to investigate the stability of the motion, we superpose upon it a two-dimensional disturbance u, v, where u and v are regarded as small. If the fluid is incompressible,
du    dv _                                            , .
and if the squares and, products of small quantities are neglected, the hydro-dynamical equations give*
"~37av~...................(2)
From (1) and (2), if we assume that as functions of t and on, u and v are proportional to ei(nt+kx}, where k is real and n may be real or complex,
(3)
^
In the paper quoted it was shown that under certain conditions n could not be complex; and it may be convenient to repeat the argument.    Let
njJc = p + iq,   v = a + i/3,
* Proceedings of London Mathematical Society, Vol. xi. p. 57 (1880); Scientific Papers, Vol. i. p. 485.    Also Lamb's Hydrodynamics,  345......... (38)